After days of scouring the web for answers, looking extensively through project config files with reference assemblies mentioned, the above-mentioned advice to use Windows 8.1 SDK to re-install a missing 4.5.1 reference assembly finally did the trick.
In my case, I was attempting to 'start without debugging' a new ASP.Net Core Web Application (.NET Core) created to follow the tutorial at: https://docs.asp.net/en/latest/tutorials/first-mvc-app/start-mvc.html
When using the SDK, I unchecked EVERYTHING accept for the reference assemblies. This was the only way I found to install 4.5.1 because other methods would fail due to the fact that I already had 4.5.2 and higher versions installed.
Kudos to this page for the answer that finally worked!
warning MSB3644: The reference assemblies for framework
".NETFramework,Version=v4.5.1" were not found. To resolve this, install
the SDK or Targeting Pack for this framework version or retarget your
application to a version of the framework for which you have the SDK
or Targeting Pack installed.